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Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles

Part of: PLA 17th European Fusion Theory Conference

Published online by Cambridge University Press:  24 May 2018

D. A. Kaltsas Open the ORCID record for D. A. Kaltsas [Opens in a new window] ,
G. N. Throumoulopoulos Open the ORCID record for G. N. Throumoulopoulos [Opens in a new window]  and
P. J. Morrison Open the ORCID record for P. J. Morrison [Opens in a new window]
D. A. Kaltsas
Affiliation:
Department of Physics, University of Ioannina, GR 451 10 Ioannina, Greece
G. N. Throumoulopoulos*
Affiliation:
Department of Physics, University of Ioannina, GR 451 10 Ioannina, Greece
P. J. Morrison
Affiliation:
Department of Physics and Institute for Fusion Studies, University of Texas, Austin, TX 78712, USA
*
†Email address for correspondence: gthroum@cc.uoi.gr
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Abstract

Hamiltonian extended magnetohydrodynamics (XMHD) is restricted to respect helical symmetry by reducing the Poisson bracket for the three-dimensional dynamics to a helically symmetric one, as an extension of the previous study for translationally symmetric XMHD (Kaltsas et al., Phys. Plasmas, vol. 24, 2017, 092504). Four families of Casimir invariants are obtained directly from the symmetric Poisson bracket and they are used to construct Energy–Casimir variational principles for deriving generalized XMHD equilibrium equations with arbitrary macroscopic flows. The system is then cast into the form of Grad–Shafranov–Bernoulli equilibrium equations. The axisymmetric and the translationally symmetric formulations can be retrieved as geometric reductions of the helically symmetric one. As special cases, the derivation of the corresponding equilibrium equations for incompressible plasmas is discussed and the helically symmetric equilibrium equations for the Hall MHD system are obtained upon neglecting electron inertia. An example of an incompressible double-Beltrami equilibrium is presented in connection with a magnetic configuration having non-planar helical magnetic axis.

Keywords

plasma confinementplasma dynamicsplasma flows
Type
Research Article
Information
Journal of Plasma Physics , Volume 84 , Issue 3 , June 2018 , 745840301
Copyright
© Cambridge University Press 2018 

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References

Abdelhamid, H. M., Kawazura, Y. & Yoshida, Z. 2015 Hamiltonian formalism of extended magnetohydrodynamics. J. Phys. A 48 (23), 235502.Google Scholar
Almaguer, J. A., Hameiri, E., Herrera, J. & Holm, D. D. 1988 Lyapunov stability analysis of magnetohydrodynamic plasma equilibria with axisymmetric toroidal flow. Phys. Fluids 31 (7), 1930–1939.CrossRefGoogle Scholar
Andreussi, T., Morrison, P. J. & Pegoraro, F. 2010 MHD equilibrium variational principles with symmetry. Plasma Phys. Control. Fusion 52 (5), 055001.CrossRefGoogle Scholar
Andreussi, T., Morrison, P. J. & Pegoraro, F. 2012 Hamiltonian magnetohydrodynamics: helically symmetric formulation, casimir invariants, and equilibrium variational principles. Phys. Plasmas 19 (5), 052102.Google Scholar
Andreussi, T., Morrison, P. J. & Pegoraro, F. 2013 Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability-theory. Phys. Plasmas 20 (9), 092104.Google Scholar
Andreussi, T., Morrison, P. J. & Pegoraro, F. 2016 Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability-examples with translation symmetry. Phys. Plasmas 23 (10), 102112.Google Scholar
Andreussi, T. & Pegoraro, F. 2008 On the variational approach to axisymmetric magnetohydrodynamic equilibria. Phys. Plasmas 15 (9), 092108.Google Scholar
Barberio-Corsetti, P. 1973 Force-free helical equilibria. Plasma Phys. 15 (11), 1131.CrossRefGoogle Scholar
Bergerson, W. F., Auriemma, F., Chapman, B., Ding, W. X., Zanca, P., Brower, D. L., Innocente, P., Lin, L., Lorenzini, R. & Martines, E. 2011 Bifurcation to 3D helical magnetic equilibrium in an axisymmetric toroidal device. Phys. Rev. Lett. 107, 255001.Google Scholar
Bogoyavlenskij, O. I. 2000 Exact helically symmetric plasma equilibria. Lett. Math. Phys. 51 (4), 235–247.CrossRefGoogle Scholar
Chandrasekhar, S. & Kendall, P. C. 1957 On force-free magnetic fields. Astrophys. J. 126, 457.CrossRefGoogle Scholar
Cooper, W., Graves, J. & Sauter, O. 2011 Jet snake magnetohydrodynamic equilibria. Nucl. Fusion 51 (7), 072002.CrossRefGoogle Scholar
Cooper, W. A., Graves, J. P., Pochelon, A., Sauter, O. & Villard, L. 2010 Tokamak magnetohydrodynamic equilibrium states with axisymmetric boundary and a 3D helical core. Phys. Rev. Lett. 105, 035003.Google Scholar
Evangelias, A., Kuiroukidis, A. & Throumoulopoulos, G. N. 2018 Helically symmetric equilibria with pressure anisotropy and incompressible plasma flow. Plasma Phys. Control. Fusion 60 (2), 025005.Google Scholar
de Gouveia Dal Pino, E. M. 2005 Astrophysical jets and outflows. Adv. Space Res. 35 (5), 908–924.Google Scholar
Grasso, D., Tassi, E., Abdelhamid, H. M. & Morrison, P. J. 2017 Structure and computation of two-dimensional incompressible extended MHD. Phys. Plasmas 24 (1), 012110.Google Scholar
Guazzotto, L. & Betti, R. 2015 Two-fluid equilibrium with flow: Flow2. Phys. Plasmas 22 (9), 092503.Google Scholar
Hameiri, E. 2013 Ertel’s vorticity theorem and new flux surfaces in multi-fluid plasmas. Phys. Plasmas 20 (9), 092503.CrossRefGoogle Scholar
Hazeltine, R. D., Hsu, C. T. & Morrison, P. J. 1987 Hamiltonian four-field model for nonlinear tokamak dynamics. Phys. Fluids 30 (10), 3204–3211.Google Scholar
Helander, P., Beidler, C. D., Bird, T. M., Drevlak, M., Feng, Y., Hatzky, R., Jenko, F., Kleiber, R., Proll, J. H. E., Turkin, Y. et al. 2012 Stellarator and tokamak plasmas: a comparison. Plasma Phys. Control. Fusion 54 (12), 124009.Google Scholar
Holm, D. D. 1987 Hall magnetohydrodynamics: conservation laws and Lyapunov stability. Phys. Fluids 30 (5), 1310–1322.CrossRefGoogle Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123 (1), 1–116.Google Scholar
Iqbal, M., Mirza, A. M., Murtaza, G. & Yoshida, Z. 2001 High $\unicode[STIX]{x1D6FD}$ relaxed states with internal conductor plasma configuration. Phys. Plasmas 8 (5), 1559–1564.CrossRefGoogle Scholar
Johnson, J. L., Oberman, C. R., Kulsrud, R. M. & Frieman, E. A. 1958 Some stable hydromagnetic equilibria. Phys. Fluids 1 (4), 281–296.Google Scholar
Kaltsas, D. A., Throumoulopoulos, G. N. & Morrison, P. J. 2017 Translationally symmetric extended mhd via Hamiltonian reduction: Energy–Casimir equilibria. Phys. Plasmas 24 (9), 092504.Google Scholar
Kawazura, Y., Miloshevich, G. & Morrison, P. J. 2017 Action principles for relativistic extended magnetohydrodynamics: a unified theory of magnetofluid models. Phys. Plasmas 24 (2), 022103.CrossRefGoogle Scholar
Kimura, K. & Morrison, P. J. 2014 On energy conservation in extended magnetohydrodynamics. Phys. Plasmas 21 (8), 082101.CrossRefGoogle Scholar
Kruskal, M. D. & Oberman, C. R. 1958 On the stability of plasma in static equilibrium. Phys. Fluids 1 (4), 275–280.CrossRefGoogle Scholar
Lighthill, M. J. 1960 Studies on magneto-hydrodynamic waves and other anisotropic wave motions. Phil. Trans. R. Soc. Lond. A 252 (1014), 397–430.Google Scholar
Lingam, M., Morrison, P. & Tassi, E. 2015a Inertial magnetohydrodynamics. Phys. Lett. A 379 (6), 570–576.Google Scholar
Lingam, M., Morrison, P. J. & Miloshevich, G. 2015b Remarkable connections between extended magnetohydrodynamics models. Phys. Plasmas 22 (7), 072111.Google Scholar
Lorenzini, R., Martines, E., Piovesan, P., Terranova, D., Zanca, P., Zuin, M., Alfier, A., Bonfiglio, D., Bonomo, F., Canton, A. et al. 2009 Self-organized helical equilibria as a new paradigm for ohmically heated fusion plasmas. Nat. Phys. 5, 570–574.Google Scholar
Lüst, R. 1959 Über die ausbreitung von wellen in einem plasma. Fortschr. Phys. 7 (9), 503–558.Google Scholar
Mahajan, S. M. & Yoshida, Z. 1998 Double curl beltrami flow: diamagnetic structures. Phys. Rev. Lett. 81, 4863–4866.Google Scholar
Moawad, S., Ramadan, A., Ibrahim, D., El-Kalaawy, O. & Hussain, E. 2017 Linear stability of certain translationally symmetric mhd equilibria with incompressible flow. Results in Physics 7, 2159–2171.CrossRefGoogle Scholar
Moawad, S. M. 2013 Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field. J. Plasma Phys. 79 (5), 873–883.Google Scholar
Morrison, P. J. 1982 Poisson brackets for fluids and plasmas. AIP Conf. Proc. 88 (1), 13–46.Google Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467–521.CrossRefGoogle Scholar
Morrison, P. J. & Greene, J. M. 1980 Noncanonical hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics. Phys. Rev. Lett. 45, 790–794.Google Scholar
Morrison, P. J., Lingam, M. & Acevedo, R. 2014 Hamiltonian and action formalisms for two-dimensional gyroviscous magnetohydrodynamics. Phys. Plasmas 21 (8), 082102.CrossRefGoogle Scholar
Morrison, P. J., Tassi, E. & Tronko, N. 2013 Stability of compressible reduced magnetohydrodynamic equilibria-analogy with magnetorotational instability. Phys. Plasmas 20 (4), 042109.Google Scholar
Pecquet, A. L., Cristofani, P., Mattioli, M., Garbet, X., Laurent, L., Geraud, A., Gil, C., Joffrin, E. & Sabot, R. 1997 Snake-like phenomena in Tore Supra following pellet injection. Nucl. Fusion 37 (4), 451.Google Scholar
Pudritz, R., Hardcastle, M. & Gabuzda, D. 2012 Magnetic fields in astrophysical jets: from launch to termination. Space Sci. Rev. 169 (1–4), 27–72.CrossRefGoogle Scholar
Puiatti, M. E., Alfier, A., Auriemma, F., Capello, S., Carraro, L., Cavazzana, R., Dal Bello, S., Fassina, A., Escande, D. F., Franz, P. et al. 2009 Helical equilibria and magnetic structures in the reversed field pinch and analogies to the tokamak and stellarator. Plasma Phys. Control. Fusion 51 (12), 124031.CrossRefGoogle Scholar
Spitzer, L. 1958 The stellarator concept. Phys. Fluids 1 (4), 253–264.CrossRefGoogle Scholar
Tassi, E., Morrison, P. J., Waelbroeck, F. L. & Grasso, D. 2008 Hamiltonian formulation and analysis of a collisionless fluid reconnection model. Plasma Phys. Control. Fusion 50 (8), 085014.Google Scholar
Terranova, D., Bonfiglio, D., Boozer, A. H., Cooper, A. W., Gobbin, M., Hirshman, S. P., Lorenzini, R., Marrelli, L., Martines, E., Momo, B. et al. 2010 A 3D approach to equilibrium, stability and transport studies in RFX-mod improved regimes. Plasma Phys. Control. Fusion 52 (12), 124023.Google Scholar
Throumoulopoulos, G. N. & Tasso, H. 1999 Ideal magnetohydrodynamic equilibria with helical symmetry and incompressible flows. J. Plasma Phys. 62 (4), 449–459.Google Scholar
Throumoulopoulos, G. N. & Tasso, H. 2006 On hall magnetohydrodynamics equilibria. Phys. Plasmas 13 (10), 102504.CrossRefGoogle Scholar
Tsinganos, K. C. 1982 Magnetohydrodynamic equilibrium. III – helically symmetric fields. Astrophys. J. 259, 820–831.CrossRefGoogle Scholar
Uo, K. 1961 The confinement of plasma by the heliotron magnetic field. J. Phys. Soc. Japan 16 (7), 1380–1395.Google Scholar
Waelbroeck, F. 2009 Theory and observations of magnetic islands. Nucl. Fusion 49 (10), 104025.Google Scholar
Weller, A., Cheetham, A. D., Edwards, A. W., Gill, R. D., Gondhalekar, A., Granetz, R. S., Snipes, J. & Wesson, J. A. 1987 Persistent density perturbations at rational-q surfaces following pellet injection in the joint European torus. Phys. Rev. Lett. 59, 2303–2306.Google Scholar
Yoshida, Z. & Hameiri, E. 2013 Canonical Hamiltonian mechanics of hall magnetohydrodynamics and its limit to ideal magnetohydrodynamics. J. Phys. A 46 (33), 335502.Google Scholar
Yoshida, Z. & Mahajan, S. M. 2002 Variational principles and self-organization in two-fluid plasmas. Phys. Rev. Lett. 88, 095001.Google Scholar
Yoshida, Z., Mahajan, S. M., Ohsaki, S., Iqbal, M. & Shatashvili, N. 2001 Beltrami fields in plasmas: high-confinement mode boundary layers and high beta equilibria. Phys. Plasmas 8 (5), 2125–2131.Google Scholar
Yoshida, Z., Morrison, P. & Dobarro, F. J. 2014 Singular casimir elements of the Euler equation and equilibrium points. J. Math. Fluid Mech. 16 (1), 41–57.CrossRefGoogle Scholar